Theorem bnj1071 35374

Description: Technical lemma for bnj69 35407. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 35166	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7868	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6375	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 19	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ∖ cdif 3901  ∅c0 4285  {csn 4588   E cep 5559   Fr wfr 5610  Ord word 6359  ωcom 7860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-ss 3921  df-uni 4872  df-tr 5218  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-om 7861

This theorem is used by:  bnj1030  35384  bnj1133  35386